Basic

A chain complexV•V_\bullet is a sequence {Vn}n∈ℤ\{V_n\}_{n \in \mathbb{Z}} of abelian groups or modules (for instance vector spaces) or similar equipped with linear maps {dn:Vn+1→Vn}\{d_n : V_{n+1} \to V_n\} such that d2=0d^2 = 0, i.e. the composite of two consecutive such maps is the zero morphismdn∘dn+1=0d_n \circ d_{n+1} = 0.

A basic example is the singular chain complex of a topological space, or the de Rham complex of a smooth manifold. Another type of example occurs with the Dold-Kan correspondence as the Moore complex of a simplicial abelian group or similar. Both the first and the third of these types of example correspond, on the surface, to chain complexes in which the grading is by ℕ\mathbb{N}, not ℤ\mathbb{Z}. Dually the de Rham complex example can be included by indexing by the non-positive integers, but by defining them to take trivial, that is zero, values in other dimensions they become chain complexes in the sense used here. The more general definition is important as it is (i) more inclusive and (ii) leads to objects that behave well with respect to shift / translation operators, (see below).

In particular this reasoning shows that connective chain complexes of abelian groups are a model for abelian ∞-groups (aka Kan complexes with abelian group structure). This correspondence figures as part of the cosmic cube-classification scheme of n-categories.

Definition

A cochain complex in 𝒞\mathcal{C} is a chain complex in the opposite category𝒞op\mathcal{C}^{op}. Hence a tower of objects and morphisms as above, but with each differentialdn:Vn→Vn+1d_n : V^n \to V^{n+1} increasing the degree.

Remark

One also says homologically graded complex, for the case that the differentials lower degree, and cohomologically graded complex for the case where they raise degree.

Remark

Frequently one also considers ℕ\mathbb{N}-graded (or nonnegatively graded) chain complexes (for instance in the Dold-Kan correspondence), which can be identified with ℤ\mathbb{Z}-graded ones for which Vn=0V_n=0 when n<0n\lt 0. Similarly, an ℕ\mathbb{N}-graded cochain complex is a cochain complex for which Vn=0V_n=0 when n<0n\lt 0, or equivalently a chain complex for which Vn=0V_n=0 when n>0n\gt 0.

In terms of translations

Note that in particular, a chain complex is a graded object with extra structure. This extra structure can be codified as a map of graded objects ∂:V→TV\partial:V\to T V, where TT is the ‘shift’ endofunctor of the category Gr(V)Gr(V) of graded objects in CC, such that T(∂)∘∂=0T(\partial) \circ \partial = 0. More generally, in any pre-additive category GGwith translationT:G→GT : G \to G, we can define a chain complex to be a differential object∂V:V→TV\partial_V : V \to T V such that V→∂VTV→T(∂V)TTVV \stackrel{\partial_V}{\to} T V \stackrel{T(\partial_V)}{\to} T T V is the zero morphism. When G=Gr(C)G= Gr(C) this recovers the original definition.